New Basis Functions for Model Reduction of Nonlinear PDEs

The selection of the spatial basis functions is very important for model reduction of the nonlinear partial differential equations (PDEs) under time/space separation framework, which will significantly affect the accuracy and efficiency of the modeling. Using the spatial basis functions expansions and the Galerkin method, the finite-dimensional ordinary differential equation (ODE) systems can be obtained from the PDEs. However, the general basis functions are not optimal in the sense that the dimensions of the ODE system are not lowest at a given modeling accuracy. The current study proposes new basis functions for the model reduction of nonlinear PDEs, which are obtained by linear combinations of general spatial basis functions. The transformation matrix for the combination coefficients is derived from straightforward optimization techniques for an improved spatio-temporal error function between the approximation and the measured spatial-temporal output. The derivation of the improved error functions also considers the influence of the variance of the spatialtemporal error. Using the new basis functions expansions and Galerkin method, it can provide a lower dimensional and more precise ODE to approximate the dynamics of the nonlinear PDEs. The modeling performance are compared with the method proposed in reference, and the simulations shows the feasibility and effectiveness of the proposed new basis functions for model reduction of nonlinear PDEs.